Polymer translocation through a nanopore under an applied external fieldKaifu Luo, Ilkka Huopaniemi, Tapio Ala-Nissila, and See-Chen Ying

Citation: The Journal of Chemical Physics 124, 114704 (2006); doi: 10.1063/1.2179792 View online: http://dx.doi.org/10.1063/1.2179792 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/124/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Simulation study on the translocation of a partially charged polymer through a nanopore J. Chem. Phys. 137, 034903 (2012); 10.1063/1.4737929 Hydrodynamic effects on the translocation rate of a polymer through a pore J. Chem. Phys. 131, 044904 (2009); 10.1063/1.3184798 Effect of attractive polymer-pore interactions on translocation dynamics J. Chem. Phys. 130, 054902 (2009); 10.1063/1.3071198 Polymer translocation through a nanopore induced by adsorption: Monte Carlo simulation of a coarse-grainedmodel J. Chem. Phys. 121, 6042 (2004); 10.1063/1.1785776 Polymer translocation through a nanopore. II. Excluded volume effect J. Chem. Phys. 120, 3460 (2004); 10.1063/1.1642588

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Polymer translocation through a nanopore under an applied external fieldKaifu Luoa� and Ilkka HuopaniemiLaboratory of Physics, Helsinski University of Technology, P.O. Box 1100, FIN-02015 HUT, Espoo, Finland

Tapio Ala-Nissilab�

Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, Espoo, Finlandand Department of Physics, Brown University, Providence, Box 1843, Rhode Island 02912-1843

See-Chen Yingc�

Department of Physics, Brown University, Providence, Box 1843, Rhode Island 02912-1843

�Received 15 November 2005; accepted 31 January 2006; published online 15 March 2006�

We investigate the dynamics of polymer translocation through a nanopore under an externallyapplied field using the two-dimensional fluctuating bond model with single-segment Monte Carlomoves. We concentrate on the influence of the field strength E, length of the chain N, and length ofthe pore L on forced translocation. As our main result, we find a crossover scaling for thetranslocation time � with the chain length from ��N2� for relatively short polymers to ��N1+� forlonger chains, where � is the Flory exponent. We demonstrate that this crossover is due to thechange in the dependence of the translocation velocity v on the chain length. For relatively shortchains v�N−�, which crosses over to v�N−1 for long polymers. The reason for this is that withincreasing N there is a high density of segments near the exit of the pore, which slows down thetranslocation process due to slow relaxation of the chain. For the case of a long nanopore for whichR�, the radius of gyration Rg along the pore, is smaller than the pore length, we find no clear scalingof the translocation time with the chain length. For large N, however, the asymptotic scaling ��N1+� is recovered. In this regime, � is almost independent of L. We have previously found that fora polymer, which is initially placed in the middle of the pore, there is a minimum in the escape timefor R� �L. We show here that this minimum persists for weak fields E such that EL is less than somecritical value, but vanishes for large values of EL. © 2006 American Institute of Physics.�DOI: 10.1063/1.2179792�

I. INTRODUCTION

Many crucially important processes in biology involvethe translocation of a biopolymer through nanometer-scalepores, such as DNA and RNA translocations across nuclearpores, protein transport through membrane channels, and vi-rus injection.1–3 Due to various potential technological appli-cations, such as rapid DNA sequencing,4,5 gene therapy, andcontrolled drug delivery,6 polymer translocation has been thesubject of a number of experimental,7–17 theoretical,17–35 andnumerical studies.32–43 In order to overcome a large entropicbarrier typical to polymer translocation and to speed up thetranslocation, an external driving force is needed, such as anelectric field, chemical potential difference, or selective ad-sorption on one side of the membrane. There have also beenseveral theoretical studies on chain translocation in the pres-ence of binding particles.28,29,43

The nanopore detection and analysis of single moleculesis based on the working principle of a Coulter counter.44

When a particle passes through a nanopore, the electrolyte inthe solution is displaced, resulting in blockades in the ioniccurrent. The magnitude of these blockades is roughly propor-

a�Author to whom correspondence should be addressed. Electronic mail:luokaifu@yahoo.com

b�Electronic mail: tapio.ala-nissila@tkk.fic�

Electronic mail: ying@physics.brown.edu

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tional to the volume of the particle. Kasianowicz et al.7 dem-onstrated that an electric field can drive single-stranded DNA�ssDNA� and RNA molecules through the water-filled�-hemolysin channel and that the passage of each moleculeis signaled by a blockade in the channel current. The trans-location process includes two essential steps. First, one endof the polymer enters the pore directed by diffusion and bythe action of an electric field near the pore. Second, the poly-mer is translocated from one side of the membrane to theother, driven by the electric field. For the first step, the ex-perimental results show that the ability of the polymer toenter the nanopore depends linearly on polymerconcentration.7,10 For the second step, the translocation timeis highly sensitive to the polynucleotide sequence of ssDNAand RNA and secondary structure of RNA.8 As to the depen-dence of the polymer translocation on the chain length, tworegimes are found, depending on the polymer length.11 Forlong polymers, the mean translocation time appears to belinear with the chain length,7,11 while it decreases rapidlywith decreasing chain length in a nonlinear way.11 In addi-tion, an inverse linear dependence and an inverse quadraticdependence of the translocation time on applied voltage areobserved for different experiments.7,11

Only a limited voltage range can be applied across abiological pore. Furthermore, there are difficulties in analyz-

ing the current variations because the shot noise is compa-

© 2006 American Institute of Physics04-1

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rable to the expected signal. Recently, solid-state nanoporeshave been used for similar experiments.14–17 Storm et al.17

carried out a set of experiments on double-stranded DNA�dsDNA� molecules with various lengths that translocatethrough a solid-state nanopore. Surprisingly, a power-lawscaling of the most probable translocation time with thepolymer length was observed with an exponent of 1.27, incontrast to the linear behavior observed for the experimentson �-hemolysin channel.7,11

Thus existing theories17,19,22,23 provide different predic-tions for the scaling behavior of the translocation time as afunction of polymer length, ranging from ��N to ��N1+�,where � is the Flory exponent.45,46 Moreover, they do notagree with the recent experimental finding17 that ��N1.27. Toclarify these issues, here we perform numerical simulationstudies based on a two-dimensional �2D� fluctuating bondlattice model for polymers to investigate the translocationdynamics under an external driving field within the pore. Asour main result, we find that for short nanopores, the trans-location time crosses over from N2� to N1+� with increasingchain length. In three dimensions �3D�, the exponent45,46

2�=1.18, which is in reasonably good agreement with theexperimental result of 1.27. We demonstrate that the cross-over is due to a change in the translocation velocity of thepolymer as a function of the chain length. Finally, we alsodiscuss the influence of the pore length on forced transloca-tion.

II. THE FLUCTUATING BOND MODEL

The fluctuating bond �FB� model47 with single-segmentMonte Carlo �MC� moves is an efficient model to study vari-ous static and dynamic properties of polymers. Here, we con-sider the 2D lattice FB polymer model for MC simulations ofa self-avoiding polymer, where each segment excludes fournearest and next nearest neighbor sites on a square lattice.The bond lengths bl are allowed to vary in the range 2�b1

��13 in units of the lattice constant, where the upper limitprevents bonds from crossing each other. With these restric-tions each segment can occupy 36 different lattice sites andthere are 28 different bond angles �� �0,��, thus yielding areasonable approximation for continuum behavior.

The external driving force in the present work was mod-eled as a potential difference applied linearly across thelength of the pore with the profile in the manner described byChern et al.,36

Ue

kBT=

−E

kBT

L

2, x � L/2

−E

kBTx , L/2 � x � − L/2 and y2 � �W/2�2

E

kBT

L

2, x − L/2

, otherwise,

�1�

where L and W are the length and width of the pore, respec-

tively, and E is the strength of the external field.ticle is copyrighted as indicated in the article. Reuse of AIP content is subje

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Dynamics is introduced by Metropolis moves of a singlesegment, with a probability of acceptance min�e−�Ue/kBT ,1�,where �Ue is the energy difference between the new and oldstates. As to an elementary MC move, we randomly select amonomer and attempt to move it onto an adjacent lattice site�in a randomly selected direction�. If the new position doesnot violate the excluded-volume or maximal bond-length re-strictions, the move is accepted or rejected according to Me-tropolis criterion. N elementary moves constitute one MCtime step. In the FB model, each effective monomer corre-sponds to several real chemical monomers along the back-bone of the chain. Each segment separating two adjacenteffective monomers has a physical meaning correspondingapproximately to the Kuhn length a which measures thelength scale at which the polymer is stiff and not flexible.

III. RESULTS AND DISCUSSION

To begin the simulation of translocation of the polymerthrough the pore, a chain is placed on one side of the porewith one end of it in the pore entrance. Then the chain isallowed to reach an equilibrium state using MC moves, butwith the constraint that the first monomer is fixed. Once thepolymer is in its equilibrium state, the first monomer at theentrance of the pore is released, and that moment is desig-nated as �=0. The translocation time � is defined as the du-ration of time it takes for the chain to move through the porein the direction of the driving force. In our simulations, thepore width is fixed as two lattice units unless otherwisestated.

A. Polymer translocation through a short nanopore

1. Influence of the electric field strength ontranslocation time

To determine how translocation time depends on theelectric field strength E /kBT, we considered a polymer chainof length N=75. The length and the width of the pore werechosen as 3 and 2 lattice units, respectively. It is important tonote that not all the simulation runs result in a successfultranslocation and even when they do, the translocation timesvary over a wide range of values. We define the translocationtime � as the average time over all successful runs. For thepresent case where the driving force is relatively strong, thedistribution of translocation times is narrow without a longtail and symmetric with respect to the most probable trans-location time, as noted by Kantor and Kardar.33 Therefore,the average is well defined. With increasing electric field thetranslocation time decreases rapidly for weak fields and satu-rates to a constant value for strong fields, as shown in Fig. 1.This is because the electric field interacts with the polymeronly inside the pore and therefore within our model, thetranslocation rate has a finite maximum. For weak fields, wefind that ��E−0.83±0.01, which is in agreement with the ex-periments of Kasianowicz et al.7 who found that the channelblockade lifetime was inversely proportional to the appliedvoltage. By contrast, Meller et al.11 found an inverse qua-dratic dependence of apparent translocation velocities on the

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2. Numerical results for a crossover scaling behaviorof � with N

Next, we consider the influence of the chain length ontranslocation for the short pore case �L=3�. Figure 2 showsthe dependence of � on N for two different fields. One of themain features is that there is a crossover for both cases. Fora smaller chain of length N200, ��N1.46±0.01 is observed.This exponent is close to 2�, where �=0.75 is the Floryexponent for a self-avoiding walk in 2D. For longer chains oflength N�300, the slopes become 1.70 and 1.72, which arein good agreement with �+1=1.75. With increasing thewidth of the pore from 2 to 5 lattice units, we observed thesame exponents and crossover behavior. Thus, we concludethat for smaller chain length, translocation time versus poly-mer chain length satisfies ��N2� and it crosses over to��N1+� for larger N independent of the strength of the field.A crossover of � as a function of N has also been reported invery recent Langevin dynamics simulations.48

Our results are in contrast with the experimental datathat � depends linearly on N in the case of �-hemolysin,7–11

but the predicted short chain exponent 2�=1.18 in 3D agreesreasonably well with the solid-state nanopore experiments ofStorm et al.,17 who found an exponent of 1.27. The ssDNA isa flexible polymer and the Kuhn length a�1–5 nm�2–10 bps �base pairs�,49,50 while the dsDNA is a semiflex-ible polymer and typically a�100 nm�340 bps.51,52 The

FIG. 1. The influence of the strength of the field on the average transloca-tion time for chain of length N=75.

FIG. 2. Average translocation time as a function of the polymer length N for

two different field strengths.

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beginning of the crossover region occurs at N=200, whichcorresponds to real lengths of the ssDNA and the dsDNAabout 400–2 kbps �kilo-base-pairs� and 68 kbps, respec-tively. These lengths are beyond or near the longest ssDNAand dsDNA used in the experiments so far.7,11,17 Thus, it isnot surprising that crossover in scaling behavior has not beenexperimentally observed yet.

3. Comparison with theoretical scaling predictions

A number of recent theories17–35 have been developedfor the dynamics of polymer translocation. Sung and Park19

considered equilibrium entropy of the polymer as a functionof the position of the polymer through the nanopore. Underthe instantaneous equilibrium approximation during thetranslocation process, the translocation problem is reduced tothe escape of a “particle” over an entropic barrier. The lim-iting case of an extremely long chain, an infinitely thin mem-brane, and narrow pore was considered. For a Gaussian chainunder an external force, it is shown that ��N /D, where D isthe relevant diffusion coefficient. Muthukumar22 suggestedthat D is not that of the whole chain, but rather the diffusioncoefficient of the monomer that just passes the pore, andhence it is a constant independent of N. As a result, a lineardependence ��N is obtained under a strong field. This is inagreement with some experimental results7,11 for polymertranslocation through �-hemolysin channel. In addition,there is support for the linear scaling behavior from the 3DGaussian chain MC simulations of Chern et al.36 and Lange-vin dynamics simulations for relatively short polymers.37,42

However, the above theories cannot explain the recent ex-perimental result, namely, that ��N1.27 for polymer translo-cation through a solid-state nanopore.17 Incidentally, accord-ing to our present results in the short chain limit, � scales asN2� which would also yield a linear scaling result for theGaussian chain since 2�=1 in this case.

Chuang et al.32 and Kantor and Kardar33 have arguedthat the assumption of equilibrium in Brownian polymer dy-namics by Sung and Park,19 and Muthukumar22 breaks downwhen the translocation time is shorter than the equilibrationtime for the polymer, which occurs for a self-avoiding poly-mer in the long chain limit. For a Gaussian polymer, theequilibrium assumption is marginal even in the absence of adriving field. Instead, a lower bound for translocation timewas obtained by considering the unimpeded motion of apolymer that best mimics the situation with a chemical po-tential difference applied across the pore.33 Absent the re-striction imposed by the membrane, there is a force that isapplied to a single monomer, at the spatial position where themembrane would reside. Because the monomer to which theforce is applied changes constantly, it was assumed that thereis no incentive for a drastic change in the shape of the poly-mer and the scaling of the size remains the same, indepen-dent of ��. At each moment a force is applied to the poly-mer. The drift velocity u of the polymer due to the appliedforce F can be written in the scaling form as32 u�F���Rg /�r���FRg /kBT��N−�1+����FN��, where Rg is the ra-dius of gyration which scales as Rg�N�, and �r is the relax-ation time of the polymer that scales as �r�N1+2� for Rouse

dynamics. If u is proportional to the force, the scaling func-ct to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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tion � is linear and thus the mobility u /F scales as 1 /N.They thus conclude that the time for such unhindered motionscales as33

����� �Rg

u�

N1+�

��. �2�

This provides a lower bound for the scaling of the transloca-tion time which agrees with our numerical results for largeN. The fact that the numerical study of Kantor and Kardardid not yield the scaling form N1+� was attributed to the factthat the large N limit has not been reached yet.33

For the shorter polymers, the situation under an electricfield driving force is different. In this case, the equilibriumradius of gyration Rg is not an appropriate variable for thescaling form, since the shape of the polymer can be greatlydistorted by the applied field. Instead, if we denote byNtrans�t� the number of segments that have passed through thepore, we can write a scaling form for dNtrans�t� /dt as sug-gested in Refs. 32 and 34:

dNtrans�t�dt

N

�r����N

kBT� N1−2��� , �3�

where �� is the chemical potential difference between twosides of the membrane. Integrating both sides from the be-ginning to the end of the translocation process yields theresult for the scaling of the translocation time as

� � N2�, �4�

in agreement with our result for short chains. We note thatthe same scaling law in Eq. �4� was recently derived byStorm et al.17 based on the force balance between the drivingforce and the hydrodynamic friction experienced by thepolymer. They thus attributed their experimentally observednonlinear scaling to hydrodynamic interaction. However, fur-ther theoretical and numerical support for their argument iscurrently missing. The effect of hydrodynamic interaction onpolymer translocation is nontrivial and will be investigatedin future work.

4. Crossover behavior of translocation time

Although the above scaling arguments support our nu-merical results for small and large N, the crossover scalingbehavior cannot be understood based on these arguments.Theoretically, we need to answer two questions: Does Rg

remain constant during the translocation under an externalfield, and is the translocation velocity really inversely pro-portional to N for a wide range of N?

To study these assumptions, we have numerically calcu-lated Rg during the translocation process for L=3 and W=2,as shown in Fig. 3. For both N=100 and N=500, duringtranslocation Rg first increases and reaches a maximum, andthen decreases with time. The same behavior was verifiedeven for a relatively wide pore with W=200. In addition,there is a slight asymmetry of Rg with time in that it issomewhat larger before the translocation than immediatelyafter it. This indicates that the chain remains in a nonequi-

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change in the shape of the polymer during the translocationis invalid.

To study this issue in detail, we calculated the averageinitial horizontal distance between the last monomer of thepolymer and the wall R0, as shown in Fig. 4. According tothe definition of the translocation time, at the completion ofthe translocation process, the chain has moved a distance ofR0 along the direction parallel to the axis of the pore. Herewe should point out that R0 is the component of an end-tethered chain along the pore and we thus have R0��N�. Wenow define the translocation velocity as53 v= R0� /�. For a

FIG. 3. Radius of gyration of the polymer during the translocation for �a�N=100 and �b� N=500.

FIG. 4. A schematic figure showing the definition of R0.

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crossover scaling behavior of � with N, the scaling of v withN must have a crossover, too. According to our numericalresults, we must have that

v �N�

��

N�

N2� � N−� for small N

N�

N1+� � N−1 for large N . �5�

Figure 5 shows the influence of the chain length on the trans-location velocity. As expected, there is a crossover. For N�200, we find v�N−0.97, where the exponent is indeed closeto −1. For N200, we find v�N−0.63. This is in reasonablygood agreement with −�=−0.75.

The reason why the translocation velocity slows downfor large N can be seen in Fig. 6, which shows the chainconfigurations for N=100 and 600 just at the moment aftertranslocation. The density of segments near the pore is muchhigher for a long chain than for a short chain. At the latestages of translocation, the high density of segments near thepore slows down the translocation velocity. The translocationtime is much shorter than the Rouse relaxation time for aself-avoiding chain, and thus Fig. 6 demonstrates the factthat the polymer remains in a nonequilibrium state, as thetranslocated segments do not have enough time to diffuseaway from the vicinity of the pore.

5. Waiting times for monomers

An important issue from the experimental point of viewconcerns the dynamics of single monomers passing throughthe nanopore during translocation. The nonequilibrium na-ture of the driven translocation problem should have a sig-nificant effect on this. To this end, we numerically calculatedthe average waiting time for each monomer to pass throughthe pore. This is defined as the time duration between theevents when monomers s and s+1 exit the pore. In Fig. 7 weshow the results for two chain lengths. There is strong de-pendence of the waiting time on the position of the monomerin the chain. For N=100, the longest waiting time approxi-mately corresponds to the middle monomer of the chain.However, for N=400, approximately the 300th monomerneeds the longest time to thread the pore, which indicates

FIG. 5. Translocation velocity as a function of the chain length.

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segments of a long polymer near the pore slows down thetranslocation. For sequencing DNA, one expects to distin-guish monomers one by one according to the blockade ofcurrent and the waiting time. From our numerical results,even for identical monomers, the waiting times are differentand are determined by the monomer positions in the chain.For heteropolymers, it thus becomes very difficult to distin-guish based on the waiting time how many monomers of thesame kind are connected together in the chain, which is veryimportant for successful sequencing DNA.

B. Polymer translocation through a long nanopore

In this section, we discuss the influence of the porelength on translocation dynamics. Here we fixed the fieldstrength to be E /kBT=5, which means that the voltage dropincreases with increasing the pore length. Figure 8�a� showsthe translocation time as a function of the chain length fordifferent pore lengths L. For R� /L�1, where R� is the radiusof gyration along the pore, the asymptotic scaling ��N1+� isrecovered for all L. However, for a smaller ratio of R� /L thesituation is more complicated. For L=6, scaling follows theprevious short pore result ��N2� as expected, while forlonger pores there is no obvious power-law scaling. Our dataalso indicate that translocation times decrease rapidly withdecreasing chain length in a nonlinear way for short poly-mers, which is in good agreement with the experimental

11

FIG. 6. Typical polymer configurations at the moment after translocation for�a� N=100 and �b� N=600.

results. The dependence of � on L is shown in Fig. 8�b�.ct to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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For relatively short polymers, � increases with increasing L,while it is independent of L for long polymers. On one hand,with increasing L the voltage drop increases, which leads toa faster translocation. On the other hand, with increasing Lthe polymer needs to move a longer distance, which resultsin a longer translocation time. For long enough polymers, thecancellation of these two factors leads to the lack of depen-dence of � on L.

Next, we fixed EL /2kBT=2, which means that the volt-age drop across the pore does not change with the length ofthe pore. Figure 9 shows � as a function of the pore length Lfor different chain lengths. It can be seen that � increaseswith increasing L. This is because the field decreases and thedistance that the polymer has to move increases with increas-ing L.

In our previous work,35 in the absence of an externaldriving force we considered a polymer which is initiallyplaced in the middle of the pore and studied the escape time�e required for the polymer to completely exit the pore oneither end. We showed that �e has a minimum as a function ofL when the radius of gyration along the pore R� �L. To studywhether this still holds for the forced case, we investigatedthe influence of driving on �e as a function of L with constantvoltage drop, as shown in Fig. 10. The width of the pore isW=7 here. For a weak field of EL /2kBT=0.05, we still ob-serve the “optimal” pore length for minimum passage time,in agreement with our previous results35 for EL /2kBT=0.

FIG. 7. The average waiting time of all segments s in the chain for �a� N=100 and �b� N=400.

However, the minimum in the escape time rapidly vanishesticle is copyrighted as indicated in the article. Reuse of AIP content is subje

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with increasing field such that for EL /2kBT=0.1, �e alreadychanges into a monotonously increasing function of L.

IV. CONCLUSION

In this work, we have investigated the problem of poly-mer translocation through a nanopore under an electric fieldbased on the 2D fluctuating bond model with single-segmentMonte Carlo moves. We examined the influence of the fieldstrength, chain length, and pore length on translocationdynamics. As our main result, we have found a crossover

FIG. 8. �a� The average translocation time as a function of chain length fordifferent pore lengths. �b� The average translocation time as a function ofpore length L for different chain lengths. The field strength is fixed asE /kBT=5, which means that the voltage drop increases with increasing porelength.

FIG. 9. The average translocation time as a function of pore length L fordifferent chain lengths. The field is fixed as EL /2kBT=2, which means that

the voltage drop does not change.

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scaling for the translocation time with chain length from��N2� for relatively short polymers to ��N1+� for longerpolymers. With increasing N, there is a high density of seg-ments near the exist of the pore due to slow relaxation of thechain, which slows down the translocation process. We dem-onstrated that the change in the dependence of the transloca-tion velocity v on the chain length determines this crossoverbehavior. For relatively short polymers v�N−�, whichcrosses over to v�N−1 for long chains. In addition, we alsoexamined the translocation through a long pore for constantfield strength. For short polymers, i.e., where Rg along thepore direction is less than the pore length, translocation timesdecrease rapidly with decreasing chain length in a nontrivialway. In the long polymer limit, the scaling relation ��N1+�

is recovered. Finally, we have also shown that for a polymerwhich is initially placed in the middle of the pore, thereexists a minimum in the escape time, which occurs at R�

�L, provided that the applied field is sufficiently weak suchthat EL is less than some critical value. For larger EL, theescape time becomes a monotonously increasing function ofthe pore length.

ACKNOWLEDGMENTS

This work has been supported in part by a Center ofExcellence grant from the Academy of Finland. We also wishto thank the Center for Scientific Computing Ltd. for alloca-tion of computer time.

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ct to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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